# Distribution of between-group sum of squares (SSB)?

Consider a standard multivariate random effects model

$$\mathbf{X}_{ij} = \boldsymbol{\mu} + \boldsymbol{\alpha}_i +\boldsymbol{\varepsilon}_{ij}$$

for $$i = 1,\ldots,m$$ and $$j = 1,\ldots,n_i$$, and where $$\boldsymbol{\alpha}_i \sim \mathcal{N}_p( \mathbf{0},\Sigma_B)$$ and $$\boldsymbol{\varepsilon}_{ij} \sim \mathcal{N}_p( \mathbf{0},\Sigma_W)$$.

Then consider that $$m$$ samples of variable sizes $$n_1, \ldots,n_m$$ are taken. One can break down the total sum of squares as: $$\underbrace{\sum_{i = 1}^m \sum_{j = 1}^{n_i} (\mathbf{x}_{ij}-\bar{\bar{\mathbf{x}}})(\mathbf{x}_{ij}-\bar{\bar{\mathbf{x}}})'}_{\mathbf{SST}} = \underbrace{\sum_{i = 1}^m \sum_{j = 1}^{n_i} (\mathbf{x}_{ij}-\bar{\mathbf{x}}_{i.})(\mathbf{x}_{ij}-\bar{\mathbf{x}}_{i.})'}_{\mathbf{SSW}} + \underbrace{\vphantom{\sum_{j = 1}^{n_i} }\sum_{i = 1}^m n_i (\overline{\mathbf{x}}_{i.}-\bar{\bar{\mathbf{x}}})(\bar{\mathbf{x}}_{i.}.-\bar{\bar{\mathbf{x}}})'}_{\mathbf{SSB}}$$ Are there any results on the distribution of the $$\mathbf{SSB}$$ matrix - whether exact or approximations?

I have not been able to find any results with this setup, and I don't really know where to start. Any help is appreciated.